\[ b_{A}(0 ; 3), B(4 ; 3),((5 ;-1) \] \( D(1 ;-1) \), \( A B C D \) is a Quad 61. Detamine the equation of AD an (1) 6.2. Show that AD\|RC 63. (alculate the length a AD and BC

The equation of line \(AD\) can be determined using the coordinates of points \(A(0, 3)\) and \(D(1, -1)\). The slope \(m\) of line \(AD\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 3}{1 - 0} = -4\). Using the point-slope form of a line equation, the equation for line \(AD\) is \(y - 3 = -4(x - 0)\), which simplifies to \(y = -4x + 3\). To show that lines \(AD\) and \(RC\) are parallel, we need the slope of line \(RC\). Assuming point \(C\) is \(B(4, 3)\) and \(R\) is \(C(5, -1)\), the slope of \(RC\) is \(m_{RC} = \frac{-1 - 3}{5 - 4} = -4\). Since both lines have the same slope of \(-4\), it confirms that \(AD\) and \(RC\) are parallel. Next, to calculate the lengths of \(AD\) and \(BC\), we apply the distance formula. For \(AD\), length \(l_{AD} = \sqrt{(1 - 0)^2 + (-1 - 3)^2} = \sqrt{1 + 16} = \sqrt{17}\). For segment \(BC\), length \(l_{BC} = \sqrt{(5 - 4)^2 + (-1 - 3)^2} = \sqrt{1 + 16} = \sqrt{17}\). Thus, both lengths are equal, measuring \(\sqrt{17}\) units each.